Received September 21, 2020, accepted October 12, 2020, date of publication October 15, 2020, date of current version October 27, 2020.
Digital Object Identifier 10.1109/ACCESS.2020.3031392
RIS-Assisted Coverage Enhancement in
Millimeter-Wave Cellular Networks
MAHYAR NEMATI , (Member, IEEE), JIHONG PARK , (Member, IEEE),
AND JINHO CHOI , (Senior Member, IEEE)
School of Information and Technology, Deakin University, Geelong, VIC 3220, Australia
Corresponding author: Mahyar Nemati (nematim@deakin.edu.au)
This work was supported by the Australian Research Council (ARC) Discovery 2020 Funding under Grant DP200100391.
ABSTRACT The use of millimeter-wave (mmWave) bandwidth is one key enabler to achieve the high data
rates in the fifth-generation (5G) cellular systems. However, mmWave signals suffer from significant path
loss due to high directivity and sensitivity to blockages, limiting its adoption within small-scale deployments.
To enhance the coverage of mmWave communication in 5G and beyond, it is promising to deploy a large
number of reconfigurable intelligent surfaces (RISs) that passively reflect mmWave signals towards desired
directions. With this motivation, in this work, we study the coverage of an RIS-assisted large-scale mmWave
cellular network using stochastic geometry, and derive the peak reflection power expression of an RIS and
the downlink signal-to-interference ratio (SIR) coverage expression in closed forms. These analytic results
clarify the effectiveness of deploying RISs in the mmWave SIR coverage enhancement, while unveiling the
major role of the density ratio between active base stations (BSs) and passive RISs. Furthermore, the results
show that deploying passive reflectors are as effective as equipping BSs with more active antennas in the
mmWave coverage enhancement. Simulation results confirm the tightness of the closed-form expressions,
corroborating our major findings based on the derived expressions.
INDEX TERMS Millimeter-wave (mmWave), reconfigurable intelligent surface (RIS), coverage, signal-tointerference ratio (SIR), stochastic geometry.
I. INTRODUCTION
Millimeter-wave (mmWave) cellular networks are widely
studied for the emerging fifth generation (5G) of mobile communication networks and beyond. The Asia-Pacific region is
supposed to give rise to the greatest share of the total contribution of mmWave communications to the gross domestic product (GDP), i.e., $212 billion, over the period 2020 to 2034 [1];
with a compound annual growth rate of 31% in the volume of
mobile data traffic [2]. These significant growths imply that
within the next decades, mmWave cellular networks will have
significantly drawn attention to deliver much higher data-rate
and capacity compared to current levels due to the availability
of wider bandwidths [3]–[5].
As a primary distinctive technical feature, the mmWave
band suffers from a higher path loss than sub-6 GHz
band. As a result, the mmWave communication range is
limited. Nevertheless, when the frequency increases, the
The associate editor coordinating the review of this manuscript and
approving it for publication was Danping He
VOLUME 8, 2020
.
wavelength decreases which results in antenna aperture
reduction. Thanks to a short wavelength (1 − 10 mm),
it is feasible to pack multiple antenna elements into limited
space at mmWave transceivers [6]. With large antenna arrays,
e.g., multiple-input multiple-output (MIMO), mmWave cellular systems can execute beamforming to provide an array
gain that compensates the frequency dependent path loss
and overcomes additional noise power [6]–[8]. However,
the mmWave communication range is still restricted due
to the mmWave propagation characteristics, e.g., scattering,
diffraction, and penetration loss [7], [8]. For instance, communications in mmWave frequencies highly suffer from penetration losses resulting in a blockage effect which mainly
affects the line-of-sight (LoS) path and non-LoS (NLoS) path
loss characteristics [6].
Wireless transmission through multiple identified paths
utilizing active MIMO relaying has been proposed as a potential solution that can reduce the blockage effect and increase
the diversity [9], [10]. However, in [11]–[13], it is shown that
full-duplex MIMO relaying has a number of drawbacks such
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
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as signal processing complexity, noise enhancement, power
consumption and self-interference cancellations at the relay
stations along with their high implementation costs. To this
end, it would be desirable to control the propagation environment in those frequencies with simple low-cost full-duplex
passive reflectors like what has been recently proposed as in
reconfigurable intelligent surfaces (RISs) [12] to mitigate
the aforementioned drawbacks.
An RIS is a software-defined metasurface containing a
large number of passive reflectors and has given rise to
the emerging ‘‘smart radio environments’’ concept [12]. The
recent advent of RISs in wireless communications enables
network operators to control the reflection characteristics of
the radio waves in an energy efficient-way [14]. The passive
reflectors in an RIS are intelligently controlled by a main
integrated circuit (IC) to adjust phase-shift of an impinging
signal. Note that the RIS design operates like a nearly-passive
surface rather than a fully passive model, since the surface
itself does not generate new RF signals, but it can be adaptively configured by a low power IC in order to beamform and
to focus the impinging RF signals towards specified direction
and locations, respectively [15]. In other words, RIS can
turn the wireless environment, which is highly probabilistic,
into a controllable and partially deterministic phenomenon
[11], [16]. It is also noteworthy that RIS mechanism
does not include any mechanical action or tilt like what
has been widely used in micro-electro-mechanical systems
(MEMS) [17], [18]. Instead, these cheap and low-complex
RIS-elements are made by electromagnetic (EM) material that are electronically controlled with a main IC and
have unique wireless communication capabilities including conventional reflect-arrays, liquid crystal surfaces, and
software-defined meta-surfaces [11], [12], [17]–[22]. Furthermore, the channel estimation in RIS-assisted systems was
investigated in [23]–[27].
In the literature, there is a significant effort to model
mmWave cellular networks under different circumstances
using stochastic geometry [6], [28]–[30]. As early works,
general stochastic geometry frameworks of mmWave cellular network were proposed in [6], [28] to model the static
objects and corresponding blockage probability using the
concept of random shape theory. Moreover, the impact of
relay on a multi-hop medium access control protocol for
60 GHz frequency was investigated in [29] when the LoS
path is blocked. In [10], [30], comprehensive coverage performance analysis of relay-assisted mmWave cellular networks
were investigated. However, the aforementioned studies did
not consider the spatial randomness of RISs deployments.
In addition, the study of impact on RIS deployment in
mmWave cellular networks is limited. In [17], [31]–[33],
general comprehensive overviews characterizing the performance of RIS-assisted communications affecting the propagation environments were provided. Moreover, in [34]–[36],
the impact of RIS deployments for non-orthogonal multiple
access (NOMA) networks were assessed. In [37], authors
investigated the RIS-aided multi-cell NOMA networks using
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a stochastic geometry model. An analytical framework for
evaluating the ergodic capacity (EC) of the RIS-assisted systems was presented in [38]. In [39], the effect of large-scale
deployment of RISs on the performance of cellular networks was studied by modeling the blockages using the line
Boolean model. In [40], an RIS-assisted MIMO framework
was proposed to randomly serve users by jointly passive
beamforming weight at the RISs and detection weight vectors
at the users. In [41], an analytical probability framework of
successful reflection of RIS for a given transmission was
provided using point processes, stochastic geometry, and random spatial processes. In [42], authors proposed a distributed
RIS-empowered communication network architecture, where
multiple source destination pairs communicate through multiple distributed RISs. In [43], an optimal linear precoder
along with an RIS deployment in a single cell for multiple
users is used to improve the coverage performance of the
communications. In [44], a characterization of the spatial
throughput for a single-cell multiuser system assisted by
multiple RISs that are randomly deployed in the cell was
provided. It showed that the RIS-assisted model outperforms
the full-duplex relay-aided counterpart system in terms of
spatial throughput when the number of RISs exceeds a certain
value. Most recently, an investigation on [45] discussed the
potential use-cases of RISs in future wireless systems using
a novel channel modeling methodology as well as a new
software tool for RIS-empowered mmWave networks.
In this paper, we aim at studying a general tractable
framework for the coverage performance of the RIS-assisted
mmWave cellular networks with a major focus on RIS and BS
densities. We use stochastic geometry as a powerful tool to
study the average signal-to-interference-ratio (SIR) behavior
over many randomly distributed BSs, RISs, and users in a
2-dimensional (2D) space. In our proposed model, BSs are
equipped with a steerable antenna array and are able to send
two beams towards a user equipment (UE). One beam is
transmitted directly towards the UE, i.e., referred to as path A;
and the other beam is sent towards the RIS and then reflected
to the UE, i.e., referred to as path B. The main contributions
of this paper are listed as follows.
• We propose a general tractable RIS-assisted approach
for SIR coverage performance in mmWave cellular networks for the first time where the message is sent by the
BS towards the UE through two different paths. We use a
diversity technique in which the system profits from the
maximum received SIR at the UE through either path A
or path B.
• Since the reflected power of passive RIS-reflectors is
largely affected by the distance between the active BS
and the RIS due to large-scale fading, we provide the
probability distribution function (PDF) of this distance
as an important quantity and discuss its dependency on
the RIS and BS densities.
• Discrete time delay values corresponding to quantized phase-shifts at each RIS-reflector is elaborated
for passive beamforming at the RIS towards the UE.
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M. Nemati et al.: RIS-Assisted Coverage Enhancement in mmWave Cellular Networks
In addition, the peak reflection power at the RIS is
assessed. It is shown that the average peak reflection
power at the RIS decreases when the active BS density
decreases. However, the reflected power reduction can
be compensated by employing RISs of a large number
of passive reflectors.
• A closed-form approximation, referred to as Approximation-I, along with a lower bound approximation,
referred to as Approximation-II, is derived for the SIR
coverage probability of the signal received by the UE
from path B, i.e., RIS-assisted path.
• Finally, we show that the RIS-assisted model provides a
great deal of flexibility to obtain a desired SIR gain. Furthermore, we show that in large BS densities when the
active BS density decreases, the co-channel interference
caused by active interferer BSs decreases faster than the
reflected power from the RIS. As a result, in large BS
densities, the decrease of active BS density improves
the SIR coverage probability in our RIS-assisted
model.
The rest of the paper is organized as follows. In Section II,
we present the system model of a baseline and RIS-assisted
mmWave cellular networks. Then, the principles of
RIS-assisted model are discussed in Section III. Subsequently, the SIR coverage analysis of the RIS-assisted model
is provided in Section IV. A comprehensive discussion on
both of the baseline and RIS-assisted models is given in
Section V. Simulation results and comparisons are presented
in Section VI. Finally, Section VII concludes the paper.
II. SYSTEM MODEL
In this section, we first present a baseline downlink mmWave
cellular network, followed by introducing the RIS-assisted
downlink mmWave network. The baseline network model
follows the standard frameworks for stochastic geometric
mmWave system analysis [46], [47], but for the reader’s
convenience, we briefly describe the basics.
We provide a set of common suppositions used for
both baseline and RIS-assisted models as follows. In a
blockage-free mmWave network, BSs are randomly located
in a 2D space according to a homogeneous Poisson point
process (PPP), denoted by 8BS , with an intensity of
λBS . Furthermore, the UEs are distributed independently
in the area, and each UE communicates with the nearest
BS to enjoy the least mean propagation loss. The probability density function (PDF) of the distance between
the UE and the nearest BS, denoted by r0 , is obtained
from the void probability in Poisson process of R2 as
follows [48]:
fr0 (r0 ) = 2πλBS r0 e−λBS πr0 .
2
(1)
It is noteworthy that so far, we consider a blockage-free
scenario. In other words, the blockage effect is not considered
in this study to simplify the analysis without loss of generality since it does not affect the conclusions of this study.
VOLUME 8, 2020
Specifically, objects that act as blockages for the communication links, e.g., buildings and trees, can be equipped with
RISs to provide indirect LoS links between the UE and the
BS and consequently the reflections enable the network to
overcome blockages as observed in [39]. Thus, we envisage
that RISs will further enhance the network coverage under
blockages, and investigating this is deferred to our future
work.
We assume that all the BSs are equipped with N isotropic
active elements for beamforming towards the targets while
the UEs are equipped with single omnidirectional antenna.
Furthermore, the recent works in [47], [49] have shown that
the empirical distribution of small-scale fading effects of the
channel and the beamforming gain of the antenna array can
be tightly approximated by using exponential distribution.
Thus, in order to maximize the mathematical tractability of
our analysis, we consider the following assumption for the
small-scale fading channel gain.
Assumption 1 (Small-Scale Channel Gain): The small-scale
fading gain is assumed to follow an exponential distribution
with mean of 1/µ. In the past, this assumption was common
in stochastic geometric coverage analysis for mathematical
tractability [6], [47], [49]–[53]. Recent works [47], [49]
revisited this exponential fading assumption, and rediscovered its feasibility even under realistic large-scale mmWave
systems, by simply tuning µ according to the mean channel
characteristics and antenna patterns. To be more specific,
authors in [47] explained that the channel gain in mmWave
systems is affected not only by the channel randomness but
also by the antenna array directions. Consequently, in [47,
Remark 5], simplified channel and radiation approximations
are provided showing that the channel gain at the typical UE
independently follow an exponential distribution, i.e., for ISO
antenna model, µ ≈ nt1nr where nt and nr are the number
of antenna elements at the transmitter and receiver sides,
respectively.
A. BASELINE mmWave CELLULAR NETWORK
Figure 1 depicts the baseline mmWave cellular network
where each BS creates one beam to send the downlink signal
towards the desired UE. The transmit signal power at the
active BSs is assumed to be constant and is denoted by
Ps . Compared to the sub-6 GHz cellular networks in [46],
in the mmWave cellular networks, beamforming is used to
converge the signal power in a specific direction towards
a desired UE due to the mmWave propagation characteristics [54]. The existing coverage analysis in [46] evaluated
the conventional cellular networks for sub-6 GHz frequency
bands. We exploit their analysis and modify it to introduce
our baseline mmWave cellular network while taking beamforming into account. In general, as shown in Figure 1, there
are two types of signal power sources. One type is the desired
signal power received by the UE from the associated BS as
the desired source, and the other type is the interference signal
power received by the UE from the interferer BSs. Based
on the uniform planar square array (UPA) in 2D space [54],
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FIGURE 2. Depiction of the RIS-assisted mmWave cellular network.
FIGURE 1. Depiction of the baseline mmWave cellular network.
the active BSs with N isotropic elements are able to create
single beam with beamwidth:
(2)
Let each interferer BS transmit with its main-lobe pointed
at a random direction. Nevertheless, as an additional gain,
it also reduces co-channel interference because the signal
from any NLOS interferer is highly attenuated [55]. Intuitively, it affects the density of interferer BSs as a modified
homogeneous PPP, denoted by 8I , with an intensity of λI as
follows:
λBS
λI = √ .
N
(3)
In other words, only a subset of interferer BSs in which their
beamforming direction covers the desired UE are considered
to be effective interferer BSs for the UE.
Let 0o denote the SIR at an independent UE in the baseline
mmWave cellular network. In this paper, we omit the noise
power in SINR and evaluate the SIR-based performance for
simplicity. Consequently, the SIR can be obtained as
Ps g0 r0−α
=
P
Ps gi ri−α
BSi ∈8I ,
i6=0
g0 r −α
P 0 −α ,
gi ri
(4)
BSi ∈8I ,
i6=0
where gi and ri are small-scale channel gain and the distance
between the ith BS, denoted by BSi , and the UE, respectively.
Here, i = 0 indicates the nearest BS which is the associated
BS and α is the path-loss exponent of large-scale fading. From
Assumption 1, we have gi ∼ exp(µ) for all i.
For the SIR coverage probability which is the probability
that the received SIR is larger than a threshold, let T denote
the threshold. Then, from (4) and similar to the analysis
in [46], the SIR coverage probability is given by
 



−α



 g0 r
(5)
Pr [0o > T ] = E Pr  P 0 −α > T  .


gi ri


BSi ∈8I
In other words, it is equivalent with the complementary cumulative distribution function (CCDF) of SIR. Eventually, after
some analysis given in Appendix VII and similar to [46], we
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Pr [0o > T ] =
2
1+ √1 T α
N
2π
ψo = √ .
N
0o =
have
1
R∞
1
α
2
T − α 1+u 2
du
.
(6)
It is noteworthy that the final coverage probability expression
is independent of the BSs’ transmit power and density. It only
depends on the beamwidth of the beams (i.e., N ), T , and α.
B. RIS-ASSISTED mmWave CELLULAR NETWORK
Suppose that there are buildings equipped with RISs in a
mmWave cellular network which are distributed based on
a homogeneous PPP, denoted by 8RIS , with an intensity of
λRIS . Each RIS has two parts: 1) passive part containing
M passive reflectors, and 2) a simple active part acting as
a phase-shift controller. Figure 2 shows the associated BS
and the UE in the coexistence of RISs and other interferer
BSs. Suppose the nearest RIS to the UE is in distances of
r1 and r2 from the associated BS and the UE, respectively.
Then, the UE communicates with the nearest RIS along with
the nearest associated BS when there is an RIS closer to
the UE than the distance between the BS and the UE, i.e.,
r2 < r0 . In other words, different from the baseline model,
here when there is an RIS closer to the UE than the distance
between the BS and the UE, i.e., r2 < r0 , the associated
BS divides its single beam into two similar beams. The first
beam is transmitted directly towards the desired UE, and
the second beam targets the nearest RIS to the UE as shown
in Figure 2. It is noteworthy that since the location of BSs
and RISs are fixed, BSs are aware of the surrounding RIS
locations. Consequently, by receiving the UE response to
the paging message broadcasted from the associated BS and
identifying its location (assuming the BS capability of passive
localization [56]), the BS can decide which RIS is closer to
the UE to split the power between two beams. As a result,
the beamwidth of each of these two beams changes, from (2)
in the baseline model, into
√
2 2π
ψs = √ ,
(7)
N
in 2D space and the transmit power of each beam at the active
BSs becomes P2s . Nevertheless, this happens only when r2 <
r0 . The PDF of r2 can be obtained from the void probability
in Poisson process of R2 as
fr2 (r2 ) = 2π λRIS r2 e−λRIS πr2 .
2
(8)
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M. Nemati et al.: RIS-Assisted Coverage Enhancement in mmWave Cellular Networks
FIGURE 3. Comparison between the baseline and RIS-assisted models
when λRIS λBS . Squares indicate RISs. In RIS-assisted model, each RIS
has its own coverage area since the UE communicates with the nearest
RIS.
With the analysis given in Appendix VII, the probability of
having an RIS within the distance between the associated BS
and the UE becomes
2
fr2 (r2 |r2 < r0 ) = 2π(λRIS + λBS )r2 e−π(λRIS +λBS )r2 . (9)
Throughout the paper, we consider λRIS λBS , as shown
in Figure 3. Since RISs are passive, they are easier and
cheaper to be implemented than the active BSs. Therefore, it satisfies the given condition of r2 < r0 and the
expression in (9) approximately becomes equivalent to (8),
i.e.,
λRIS λBS → fr2 (r2 |r2 < r0 ) ≈ fr2 (r2 ) .
(10)
Thus, in general, there are two different paths from the
BS towards the UE as shown in Figure 2. One path which is
directly from the BS to the UE, i.e., referred to as path A; and
the other path which goes through the RIS and then reflected
towards the UE, i.e., referred to as path B. Let assume each
RIS serves one UE at a time.1 The phase-shift controller at
the RIS can adjust the phase-shifts and generate a new beam
towards the UE (as described in section III-A). In fact, the RIS
acts as a passive beam-former by adjusting the phase-shifts
at the passive reflectors and converge a beam in a specific
direction. Thus, we define two states for RISs with respect
to their reflection directions as shown in Figure 4 since the
passive elements are always on and reflect the impinging
signals at all time, regardless of whether there is any UE
associated with it or not [57].
• Engaged: The nearest RIS to the UE which is engaged
for the communication assistant in path B.
• Idle: All other RISs which are not engaged in any communication are considered idle RISs. This is the default
state when the phase-controller in an idle RIS adjusts
the phase-shifts at the reflectors somehow to generate
a beam towards an empty space, e.g., sky, to avoid
1 One RIS can serve multiple UEs assuming the size of the RIS is deter-
mined based on the density of UEs around it. Hence, in high density UE
areas, a larger RIS is needed where a portion of its reflectors can be dedicated
to each UE. However, this can be considered as another resource allocation
problem which its further assessment is out of the scope of this paper.
Therefore, without loss of generality, we assume that each RIS serves one
UE at a time.
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FIGURE 4. The idle RIS reflects the signals toward the empty spaces, e.g.
Sky.
interference with the UEs. Intuitively, it is evident that
the idle RISs do not contribute to an interference.
In a nutshell, the anatomy of the communication initiation
is briefly explained as follows.
• The BS broadcasts the UE identification number as a
paging message.
• Mobile receives the paging message and identifies itself
along with its location to the BS and nearby RISs and a
successful handshake between the BS and the UE takes
place.2
• Subsequently, the BS which knows the location of the
UE and fixed RISs around it, informs the UE and its
nearest RIS which forward channel the UE has been
assigned.
• The associated RIS becomes an engaged RIS and its
phase-controller adjusts the phase-shifts at the reflectors.
Assumption 2: An engaged RIS is able to create a beam
with highly narrow beamwidth toward the UE since it contains a large number of reflectors. Moreover, the reflection
steering angle of each RIS is only limited to [0, π]. Thus,
it may receive signal only from a half of the interferer BSs.3
Intuitively, it significantly reduces the probability that the
engaged RIS contributes to an effective interference. Therefore, throughout the paper, we neglect the interference that
may be caused by engaged RIS reflections.4
In the following section, we explain the principles of the
RIS-assisted mmWave cellular networks in more details.
III. PRINCIPLES OF RIS-ASSISTED mmWave CELLULAR
NETWORK
In this section, first, we explain the phase-shift adjustment at
each passive RIS-reflector and how passive beamforming is
done by RISs. Second, a channel model for the RIS-assisted
2 It is noteworthy that angle of arrival of the UE’s response can be simply
obtained by both the BS and RIS with passive localization methods [56].
3 Besides, since the RISs are passive and their reflected power significantly
being affected by large-scale path-loss, they can only cause an interference
for nearby UEs.
4 Recently, authors in [57] showed that the interference marginally
increases by RISs deployments. However, in order to maximize the mathematical tractability, we ignore this limited interference.
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M. Nemati et al.: RIS-Assisted Coverage Enhancement in mmWave Cellular Networks
FIGURE 5. Depiction of M-passive-elements linear phased-array at the
RIS.
Consequently, the RIS executes a new passive beamforming towards the UE, i.e., in an angle of ϕ, where the maximum power of the beam, i.e., peak effective radiated power,
scales up by M 2 . In addition, in order to put the phase-shift
adjustment at each reflector into action, we may need to
ϕ
+δm might
consider implementation constraints since m` cos
c
be continues values. However, we consider discrete values
for τm that there will be a minimum phase-shift adjustment.
We quantize the continues phase-shift amplitudes by discretizing them into the implementable values.
B. BS-RIS AND RIS-UE CHANNEL MODELS
path is discussed. Third, a distribution function for the distance between the associated BS and the engaged RIS is
provided. Finally, the peak reflection power at the RIS is
assessed.
A. PHASE-SHIFT ADJUSTMENT AT RIS AND PASSIVE
BEAMFORMING
Seeing that the RIS includes M passive reflective elements,
it does not generate any transmit power by its own. The
phase controller of the RIS determines the phase-shift for
each reflected signal to create a new beam towards the UE.
Let assume that it is only able to adjust discrete phase-shifts
for the impinging signal at each passive reflector due to the
implementation constraints. In particular, in order to show
the effect of M passive reflectors of the RIS on the transmit
power, and also the phase-shift at each reflector, let simplify
the generic 2D phased-array by applying it into a 1D linear
phased-array [58] as shown in Figure 5; where the distance
between the passive reflectors is denoted by ` and distance
of the mth passive reflector to the UE is denoted by dm ,
where m = 1, · · · , M . Let s(t + δm ) denote the message
impinging the mth passive reflector at the RIS where δm
stands for the phase difference of the impinging message at
the mth reflector. Then, the reflected signal towards the UE
which is a superposition due to M passive reflectors is given
by
M
X
dm
− τm
x(t) =
s t + δm −
c
m=1
M
X
d1
m` cos ϕ
=
s t + δm −
+
− τm , (11)
c
c
m=1
where c, ϕ and τm correspond to the wave-speed, angle of the
1th element reflection toward the UE (as shown in Figure 5)
and time delay of the mth reflector. Here, the time delay,
τm , is associated with the phase-shift adjustment at the mth
reflector. In other words, τm is given by
m` cos ϕ
+ δm .
(12)
τm =
c
Therefore, the reflected signal in (11) becomes
M
X
d1
d1
x(t) =
s t−
= Ms t −
.
(13)
c
c
m=1
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In this subsection, we briefly discuss BS to RIS
(BS-RIS) and RIS to UE (RIS-UE) channel models. Due to
the passive nature of the RIS-elements, the signal transmitted
through path B suffers from double large-scale pathloss,
i.e., r1−α and r2−α ; while profiting from the passive beamforming at the RIS which causes the power of the reflected
beam scales up by M 2 . Therefore, the received power at the
UE through path B, denoted by Pu , follows
Pu ∝
M 2 λ2
Ps
×
,
2
(4π)2 r1α r2α
(14)
where λ is the wavelength. Furthermore, by taking the
small-scale fading into account for path B, the end-to-end
channel matrix H ∈ CN ×1 is given by [45]
H = C8D,
CN ×M
(15)
CM ×1
where C ∈
and D ∈
are the matrices of channel coefficients for BS-RIS and RIS-UE paths, respectively.
Additionally, 8 ∈ CM ×M represents the diagonal matrix
of phase-shift adjustment coefficients of the RIS-reflectors
which is associated with the τm discussed in the previous
subsection. It is noteworthy that the RIS needs the direction
of the UE, i.e., ϕ, in (12) to generate the matrix 8. Since
the RIS includes a large number of passive elements, it can
profit from the angle-of-arrival (AOA) localization technique
in [56] to find ϕ when UE identifies itself by broadcasting
the response to the paging message from the associated BS.
Moreover, assuming the RIS is equipped with received RF
chain, as investigated in [13], [26], the RIS can directly obtain
the channel state information (CSI) of BS-RIS and RIS-UE
paths to achieve the optimal performance.
C. DISTANCE BETWEEN THE RIS AND THE ASSOCIATED
BS
An important quantity is the distance between the engaged
RIS and the associated BS, denoted by r1 , which affects the
transmit power at the RIS considering the large-scale fading.
Although the BSs and RISs are both distributed based on
two independent homogeneous PPPs, i.e., 8BS and 8RIS , r1
depends on both r0 and r2 as shown in Figure 6. Therefore,
the distribution of r1 for given r0 and r2 equals the captured
annulus of the circumference of the circle with a radius
of r2 and origin of the UE as shown in Figure 6. Hence,
the CDF of r1 becomes the probability of the engaged RIS
VOLUME 8, 2020
M. Nemati et al.: RIS-Assisted Coverage Enhancement in mmWave Cellular Networks
FIGURE 6. The distance between the associated BS and the associated RIS
is limited to |r0 − r2 | < r1 < r0 + r2 . The red captured annulus indicates
where the potential RIS can be located when r1 6 R for given r0 and r2 .
located on annulus of 2θr2 over the whole possible area of
the circumference of 2πr2 as follows:
θ
2θr2
= . (16)
Pr (r1 < R|r0 , r2 ) = Fr1 (R|r0 , r2 ) =
2πr2
π
FIGURE 7. Comparison of the analytical and simulation results for the
PDF of r1 . λRIS = 1000 RIS2 , λBS = 25 BS2 .
km
km
From the law of Cosines in the shaded triangle,5 we have
!
2 + r 2 − R2
r
2
θ = cos−1 0
.
(17)
2r0 r2
By substituting (17) in (16) we have
Fr1 (R|r0 , r2 ) =
cos−1
r02 +r22 −R2
2r0 r2
π
(18)
Then, the PDF of r1 for given r0 and r2 can be obtained as
dFr1 (r1 )
r1
fr1 (r1 |r0 , r2 ) =
=
s
2 .
dr1
r02 +r22 −r12
πr0 r2 1 −
2r0 r2
(19)
It is noteworthy that the r1 can vary from |r0 −r2 | to r0 +r2 .
Eventually, from (1) and (8), the PDF of the r1 is given by
Z ∞ Z ∞
fr1 (r1 ) =
fr1 (r1 |r2 , r0 )f (r2 )f (r0 ) dr2 dr0 ,
r0 =0 r2 =0
s.t. |r0 − r2 | 6 r1 6 r0 + r2 .
(20)
Figure 7 shows the theoretical PDF of the r1 which coincides with the simulation results. In addition, in order to see
the impact of λRIS and λBS on the r1 , the expected value of r1
in (20) with putting its condition into the integral becomes
Z ∞ Z ∞ Z r0 +r2
E{r1 } =
r1
r0 =0 r2 =0 r1 =|r0 −r2 |
× f (r1 |r2 , r0 )f (r2 )f (r0 ) dr1 dr2 dr0 .
Thus, we have
Z ∞ Z ∞ Z r0 +r2
E{r1 } = 4π λBS λRIS
r12
r0 =0 r2 =0 r1 =|r0 −r2 |
exp −π λRIS r22 + λBS r02
dr1 dr2 dr0 .
×
s
2
2
2
2
r +r −r
1 − 0 2r02r2 1
5 a2 = b2 + c2 − 2bc cos(Â)
VOLUME 8, 2020
(21)
FIGURE 8. Impact of λRIS and λBS on E{r1 }.
This integral is numerically calculated using MATLAB as a
function of Fr1 (λRIS , λBS ) , E{r1 } which Figure 8 shows
its changes when λRIS and λBS vary. In general, an increase
in either of λRIS or λBS reduces E{r1 }. For example, for
a fixed λBS , if λRISa < λRISb , then Fr1 λRISa , λBS >
Fr1 λRISb , λBS . However, the speed of changes in Fr1
decreases when λRIS increases. Therefore, we can conclude
that from law of large numbers (LLN) theorem, if λRISa <
λRISb , λBSa < λBSb and λBSa , λBSb , λRISa , λRISb → ∞, then
Fr1 λRISa , λBSb ≈ Fr1 λRISb , λBSb in large-scale communications.
D. PEAK REFLECTION SIGNAL POWER OF RIS
The transmitted signal from the associated BS towards the
engaged RIS experiences a small-scale fading gain, denoted
by fm , while reaching at the mth RIS-reflector, i.e., from
Assumption 1, we assume fm ∼ exp(µ). Thus, the impinging
signal power at the mth passive RIS-reflector is given by
Ps
(22)
Pm = fm r1−α .
2
Let us assume an attenuation power factor for RIS-reflectors
which is denoted by β ∈ (0, 1]. Here, β is constant and
can be obtained by measuring the attenuation power of a
signal passing through the RIS-reflectors. Furthermore, since
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M. Nemati et al.: RIS-Assisted Coverage Enhancement in mmWave Cellular Networks
where ε = max (|r0 − r2 |, ) and is a minimum euclidean
distance
n ofo the BS from the RIS that prevents the divergence
of E r1−2 . Eventually, by substituting (26) and (27) in (25),
we have

 P
RIS
E

µ

2 
α

"
=
M 2 βPs
2µ2
#2
α
0
2
+ 1 FR (λRIS , λBS ) .
α
(28)
FIGURE 9. Impactof λRIS and λBS on average reflected power,
i.e., FP λRIS , λBS , E{PRIS }, M = 100, β = 1, µ = 1, α = 4, and Ps = 2 W.
the peak effective radiated power scales up by M 2 as given
in (13), the peak reflected power from the engaged RIS
towards the UE becomes
Ps
(23)
PRIS = M 2 β E{fm } r1−α .
2
Consequently the reflected power largely affected by r1−α .
With respect to (21), the average peak reflected power can be
expressed as a function of λRIS and λBS for given α, β and M
as follows
M 2 βPs
E{r1−α },
FP (λRIS , λBS ) , E{PRIS } =
(24)
2µ
which is shown in Figure 9. In other words, the dependency of
average power on λRIS and λBS comes from F̃r1 (λRIS , λBS ) ,
E{r1−α } with respect to the PDF of r1 in (20) and (21). Thus,
we conclude that in general, since α is positive, i.e., usually
α > 2 [3], the average reflected power from the RIS increases
when λRIS increases, i.e., E{r1 } decreases. For example, keep
ing λBS fixed, if λRISa < λRISb , then FP λRISa , λBS <
FP λRISb , λBS . Likewise, the reflected power from the RIS
decreases when BS density decreases as shown in Figure 9.
However, this reflected power reduction can be compensated
by employing larger RISs with more number of reflectors,
i.e., a larger M .
In order to find the SIR coverage probability of the
RIS-assisted path, i.e., path B, and simplify the analysis in
2
RIS α
which is
section IV, we introduce a raw moment of Pµ
calculated as follows:
(
2 ) 2
2 2 n o
PRIS α
M βPs α
E
=
E fmα E r1−2 . (25)
µ
2µ
Since fm ∼ exp(µ), we have
2
2
R∞
2
E fmα = fm =0 fmα µe−µfm dfm = µ− α 0 α2 + 1 ,
(26)
R∞
where 0(x) = 0 t x−1 e− t dt is the Gamma function. Moreover, from (20), we define
n o Z ∞
FR (λRIS , λBS ) , E r1−2 =
r1−2 fr1 (r1 ) dr1 , (27)
r1 =ε
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2
RIS α
Remark 1: The raw moment of Pµ
in (28) depends
on FR (λRIS , λBS ) for given α, M , µ and β. Therefore, similar to the FP (λRIS , λBS ) , E{PRIS } shown in Figure 9,
2
RIS α
is an increasing function of λRIS and λBS .
E Pµ
IV. SIR COVERAGE PROBABILITY IN RIS-ASSISTED
mmWave CELLULAR NETWORK
In order to facilitate the UE implementation for the receptions
through two paths A and B at the receiver, diversity is taken
into account. By selection diversity of two received signals,
the strongest signal is selected. Let 0A , and 0B denote the
SIRs of the received signals through paths A and B, respectively. Then, the SIR at the UE is given by
0s = max{0A , 0B }.
(29)
The SIR of the received signal from path A is similar to
the baseline model while the beamwidth and transmit √
signal
power at the active BS are being affected, i.e., ψs = 2ψo
and P2s in 2D space. Consequently, the co-channel interference is affected as an additional gain of beamforming and the
interferer BSs density becomes
r
2
λBS .
(30)
λIs =
N
Therefore, taking these changes into account and similar
to (6), the SIR coverage probability for path A can be obtained
as
1
Pr [0A > T ] = q
R
2
∞
1 + N2 T α − 2
T
α
1
α
1+u 2
.
du
(31)
On the other hand, the SIR of the received signal through
path B is given by
0B =
PRIS hr2−α
P Ps −α ,
2 gi ri
(32)
BSi ∈8I ,
i6 =0
where h denotes the small-scale fading gain between the
engaged RIS and the UE. Based on Assumption 1 we have
h ∼ exp(µ). In order to simplify the analysis to find a
closed-form expression for Pr {0B > T }, we utilize the following conversion.
Remark 2 (Power-Density Conversion): The path-loss process of r ∈ 8 with transmit power P and intensity λ is
VOLUME 8, 2020
M. Nemati et al.: RIS-Assisted Coverage Enhancement in mmWave Cellular Networks
equivalent with that of R ∈ 8̃ with a transmit power of 1
and an intensity of λ̃ given as
2
α
λ̃ = P λ.
i6=0
where
λ̃BS =
λ̃I =
λ̃RIS = E
Ps
2µ
2
α
2
Ps
2µ
(
α
λIs
(36)
2 )
α
λRIS ,
(37)
(38)
By taking the changes in (35) to (38) into account, the SIR
coverage probability in (34) becomes
!)
(
P
−α
α
.
Pr {0B > T } = E Pr h̃ > Tr2
g̃i ri
BSi ∈8̃I
(39)
Since h̃ ∼ exp(1) and Pr(h̃ > x) = 1 − Fh̃ (x) = e−x , (39)
becomes





X
g̃i ri−α 
Pr {0B > T } = E exp −Tr2α


BSi ∈8̃I


α 
 Y
r2
=E
exp −T g̃i
. (40)


ri
BSi ∈8̃I
Since g̃i ∼ exp(1), we have E e−g̃i x = [1 + x]−1 and (40)
becomes
(
)
Q
h1 iα .
(41)
Pr {0B > T } = E
r2
BSi ∈8̃I
1+T
ri
With simplification by application of
theorem [46], [54], [60],6 from (41) we have




 Y

1
h iα
E

BS ∈8̃ 1 + T r2 

ri
i
I


 Q

h
i
R
6E
f (x) = exp −λ̃I R2 (1 − f (x))dx .


VOLUME 8, 2020
υ=r0
Simplifying the integral expression in (42) is a complicated
task since the lower bound in integral is a function of r0 and
the expression inside the integral is a function of r2 which
are independent of each other. To simplify (42) and find a
closed-form expression for 0B coverage probability, at this
stage, we provide two approximations as follows.
A. APPROXIMATION-I
In this approximation, from (1) and (8), we have
s
λ̃BS
E{r0 }
E{r2,0 } =
λ̃RIS
(35)
g̃i ∼ exp(1) and h̃ ∼ exp(1).
x∈8̃I
#
!)
1
α υdυ
1−
.
1 + T rυ2
(42)
(43)
Therefore, we simply approximate
λBS
PRIS
µ
= E exp −2π λ̃I
"
∞
Z
(33)
The proof of this remark is similar to that of [59, lemma 1].
However, for the reader’s convenience, a simplified version
is provided in Appendix 65.
Therefore based on remark 2, we have 8BS → 8̃BS ,
8I → 8̃I and 8RIS → 8̃RIS ; and from (32), the SIR
coverage probability for path B is given by
 






 


−α

 Ph̃r2
>
T
, (34)
Pr {0B > T } = E Pr 

g̃i ri−α







BSi ∈8̃I ,


(
r2 ≈ ρr0 ,
where ρ =
r
λ̃BS
.
λ̃RIS
(44)
Consequently we define the following
proposition.
Proposition 1: The SIR coverage probability for path B
of a randomly located UE in RIS-assisted mmWave cellular
network with Approximation I is
Pr {0B > T } =
λ̃RIS +
2
λ̃I
(T ) α
ρ2
λ̃RIS
R∞
ρα
α
2
u=(T )− α ρ α +u 2
(45)
du
Proof: By Taking (44) into account, (42) becomes
(
#
!)
Z ∞ "
1
α υdυ
E exp −2π λ̃I
1−
(46)
r
1 + T rυ2
υ= ρ2
By employing a change of variable u =
(
2
π λ̃I (T ) α r22
E exp −
ρ2
Z
ρ2υ2
2
r22 (T ) α
ρα
∞
2
u=(T )− α
, we have
!)
α
ρα + u 2
du
(47)
By taking the average of (47) over r2 with respect to (8)
and (37), we have
Z ∞
fr2 (r2 ) exp
r2 =0
!
2
Z ∞
π λ̃I (T ) α r22
ρα
dr2 .
× −
×
α du
2
ρ2
u=(T )− α ρ α + u 2
(48)
Campbell’s
Let consider parameter J as follows
!
Z ∞
2
λ̃I
ρα
J = π λ̃RIS + 2 (T ) α
α du . (49)
2
ρ
u=(T )− α ρ α + u 2
Then, (48) becomes
Pr {0B > T } = 2π λ̃RIS
Z
∞
r2 =0
2
r2 e−J r2 dr2 =
π λ̃RIS
(50)
J
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M. Nemati et al.: RIS-Assisted Coverage Enhancement in mmWave Cellular Networks
Eventually by substituting J in (49) into (50) we have
Pr {0B > T } =
λ̃RIS +
2
λ̃I
(T ) α
ρ2
λ̃RIS
R∞
ρα
α
2
u=(T )− α ρ α +u 2
TABLE 1. System numerical parameters.
du
B. APPROXIMATION-II
In this approximation we consider a lower bound where
λRIS λBS → υ > r2 in (42). Therefore, we state the
following proposition.
Proposition 2: The lower bound SIR coverage probability
for path B of a randomly located UE in RIS-assisted mmWave
cellular network when λRIS λBS with Approximation II is
λ̃RIS
(51)
Pr 0B0 > T =
2 R∞
1
λ̃RIS +λ̃I (T ) α
α
2
u=(T )− α 1+u 2
du
Proof: Since the proof is similar to that in proposition 1,
we omit it here.
V. DISCUSSION ON BASELINE AND RIS-ASSISTED
mmWave CELLULAR NETWORKS
The SIR coverage probability of the baseline model in (6) and
path A in the RIS-assisted model in (31) does not depend on
λBS ; and the only deployment parameter to enhance the SIR
performance is N which means that with N → ∞, we have
Pr {0o > T } → 1
N →∞⇒
.
(52)
Pr {0A > T } → 1
However the lower bound SIR probability in (51) in Approximation II shows that the SIR performance for path B depends
on deployment parameters of not only N but also λBS , λRIS ,
and M . In other words, from (28) and (37), (51) can be
re-expressed as
Pr 0B0 > T
4
=
λRIS M α F1 (λBS , λRIS , α, β)
q
,
4
λRIS M α F1 (λBS , λRIS , α, β) + N2 λBS F2 (T , α)
(53)
where

h i2 
 F (λ , λ , α, β) = β α 0 2 + 1 F (λ , λ )
1 BS
RIS
R RIS
BS
µ
α
2 R∞
1

 F2 (T , α) = (T ) α
α du
−2
u=(T )
α
1+u 2
Based on remark 1, FR and consequently F1 are increasing
functions of λRIS . Therefore, we have

λRIS → ∞ 
or M → ∞ ⇒ Pr 0B0 > T → 1.
(54)

or N → ∞
Nevertheless, in practice, there are implementation issues
which might prevent these parameters to become very large.
However, there is a great deal of flexibility to select a
proper parameter for SIR enhancement. For instance, a lower
complex antenna array, i.e., smaller N , can be deployed at
188180
the BS and either higher RIS density or larger number of
reflectors M can be taken into account to provide a desired
SIR gain. On the other hand, from (53) and in large BS
densities when λBS increases, although the F1 in the numerator increases, the expression in the denominator increases
faster
than the numerator because of additive expression of
q
2
N λBS F2 (T , α). In other words, the co-channel interference
increases faster than the reflected power from the RIS when
the active BS density increases, i.e.,
Pr 0B0 > T ∝
C1 λRIS
,
C1 λRIS + C2 λBS
(55)
where C1 and C2 are assumed to be constant for specific
moments. Therefore, it is desirable to decrease the number
of active BSs (only when λBS is large and background noise
at the UE is negligible) and deploy more passive RISs. Note
that this is only applicable in large λBS where the background
noise at the UE is negligible compared to the received interference signal power. In other words, although BS density
decrement improves the SIR performance in large BS densities, the BS density cannot be too small since the performance
might be affected by the background noise at the UE. In such
a case, SIR cannot be a reliable metric and analysis should be
applied to the SINR performance which its analysis in more
details is out of the scope of this study.
VI. NUMERICAL RESULTS
In this section, the performance of the RIS-assisted is shown
by simulation results. The evaluations include: A) the SIR
coverage probability comparison for both paths A and B
in the RIS-assisted model, B) the SIR coverage probability comparison of the RIS-assisted and the baseline models, and C) the impact of λRIS and λBS on SIR coverage
probability.
We use MATLAB and the parameters used in the simulations are given in TABLE 1, unless otherwise specified.
A. SIR COVERAGE EVALUATION IN RIS-ASSISTED MODEL
Since the selection diversity of the two received signals
is taken into account in the proposed RIS-assisted model,
VOLUME 8, 2020
M. Nemati et al.: RIS-Assisted Coverage Enhancement in mmWave Cellular Networks
FIGURE 10. Coverage evaluation of the RIS-assisted model through paths
A and B. RIS densities are in RIS
, λBS = 2.5 × 10−5 BS2 , and M = 100.
2
m
m
FIGURE 11. Coverage comparison between the RIS-assisted and the
baseline models. RIS densities are in RIS
. λBS = 2.5 × 10−5 BS2 .
2
m
m
the UE is able to select the strongest signal between those
received signals through paths A and B. Figure 10 compares
the SIR coverage probabilities for both of paths A and B.
Based on (31), (45) and (51), 0A only depends on N while
0B depends on λRIS as well. Therefore, it is shown that when
, 0B is improved.
λRIS increases from 5 × 10−4 to 0.05 RIS
m2
Moreover, when λRIS becomes larger, i.e., λRIS λBS , 0B
outperforms 0A .
B. SIR COMPARISON OF RIS-ASSISTED AND BASELINE
MODELS
Figure 11 compares the the coverage performance of the
proposed model with the baseline model. It shows that
there is a minimum coverage probability when there is no
RIS-assisted path between the BS and the UE, i.e., small
λRIS results in 0s = 0A which is independent of λRIS , e.g.,
λRIS = 10−5 RIS
. This minimum coverage probability is
m2
slightly less than that of the baseline model since we have two
beams in the RIS assisted model while there is only one beam
in the baseline model with the same BS structure. In other
words, with fixed N active antennas at the BSs, the beam in
the baseline model is narrower that that of the RIS-assisted
model causes less co-channel interference. However, when
λRIS increases, the coverage probability in the RIS-assisted is
improved. In addition, bigger RISs, i.e., a larger M , performs
better than small RISs as it increases the reflected power
from the engaged RIS. On the other hand, in such a case
where there might be an infeasibility of high RIS density
deployment, larger M can be taken into account to provide
the required SIR improvement.
C. IMPACT OF λRIS AND λBS ON SIR COVERAGE
PERFORMANCE
Figure 12 shows the impact of λRIS on the proposed
RIS-assisted model. As discussed in section V, when λRIS
increases, the SIR coverage probability of 0B approaches 1.
Moreover, we can see that Approximation I is supported by
the simulation results in high λRIS because when λRIS → ∞,
VOLUME 8, 2020
FIGURE 12. Impact of λRIS on SIR coverage performance.
λBS = 2.5 × 10−5 BS2 , M = 100 and T = 5 dB.
m
the distance of r2 and its variations becomes smaller; therefore, r2 ≈ ρr0 in (44) becomes more realistic. In addition,
the SIR performances for the baseline model and the path A
in the RIS-assisted model are fixed and independent from the
λRIS variations. 0o has better performance compared to 0A
due to having a narrower beamwidth since it uses all N elements to create one beam while in the proposed model there
are two wider beams with the same BS structure. Figure 13
depicts the SIR coverage performance of the RIS-assisted and
the baseline model with respect to λBS . As λBS increases,
0o and 0A remain unchanged while 0B slightly decreases.
It shows that although higher λBS leads to a higher reflected
power from the RIS, its impact on the co-channel interference power is larger. In other words, the speed of increasing of co-channel interference power is faster than that of
reflected power from the RIS. However, it is noteworthy that
the simulation results shown in Figure 13 are for large BS
BS
densities, i.e., λBS > 25 km
2 and the background noise at
the UE is negligible compared to the received interference
power. But Figure 14 shows the results for similar simulation when SINR coverage probability is taken into account
with a noise floor of −94 dBm and small λBS values are
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M. Nemati et al.: RIS-Assisted Coverage Enhancement in mmWave Cellular Networks
system in terms of the spatial throughput as the number of
RIS-reflectors increases [44].
VII. CONCLUSION
FIGURE 13. Impact of λBS on SIR coverage performance.
λRIS = 2 × 10−2 RIS
and T = 5 dB.
2
m
FIGURE 14. SINR performance evaluation of the proposed model with
respect to the λBS when λRIS varies from 10 000 RIS2 to 20 000 RIS2 and
km
km
T = 5 dB.
shown in the logarithmic x-axis. In other words, it shows
that although in high BS densities, smaller λBS has better
performance, there is a trade-off when the λBS decreases
significantly.
Considering both simulation results in Figures 13 and 14,
we note the feasibility of our analysis in the high BS density
regime even for SINR. Nevertheless, the difference under the
low BS density regime calls for further investigation on SINR
analysis in future research.
Recent studies [44], [57] also confirm the same conclusion that the coverage performance improves after exceeding a certain ratio of RIS to BS. This critical RIS/BS ratio
threshold depends highly on the system configurations and
simulation parameters (e.g., the numbers of RIS-reflectors
and BS antennas, channel models, etc.), and investigating
on finding the specific values. Moreover, In [44], it was
shown that compared to the conventional active relaying, RIS
requires much lower hardware cost and energy consumption
due to passive reflection, and works spectral efficiently in
full-duplex without the need of costly self-interference cancellation techniques [44]. In addition, it was discussed that the
RIS-aided system outperforms the full-duplex relaying based
188182
In this study, we proposed a new RIS-assisted mmWave
cellular network where a message is transmitted by a BS
towards a desired UE though two LoS and NLoS paths.
The NLoS path passes through an RIS and then, reflected
towards the UE. Discrete time delay values corresponding the
phase-shifts at each RIS-reflector was elaborated and the peak
reflection power at the RIS was assessed. Since the UE utilizes selection diversity technique to pick the strongest signal
received through the two paths, we analysed the SIR coverage
performance of both paths with major emphasis on RIS and
BS densities and compared its performance with a baseline
model. Two closed-form approximations were derived analytically for the SIR coverage probability of the RIS-assisted
path. It was shown that the SIR coverage probability of the
RIS-assisted path depends on not only N but also λRIS , λBS ,
and M which provides a great deal of flexibility to obtain
a desired SIR gain. Furthermore, we note that the density
magnitudes seem to be large numbers. Nevertheless, the RIS
concept is supposed to be taken into action in 6G and in
this proposed network model there are some assumptions
related to the considered system model to show the general
performance of the system while practicality feasibility of
such implementations is more relevant to the commercial
value of the proposed method. Besides, it is also noteworthy
that massive MIMO concept, and recently proposed radio
stripes patented in 2017 [61] or even holographic beamforming proposed in the 90’s [62], [63] can help and support the technical and commercial implementation of such
surfaces.
APPENDIX A
DERIVATION OF Pr [0o > T]
Similar to the analysis in [46], the SIR coverage probability
in (5) becomes
Pr [0o > T ]






 

X

−α 
α

=
gi ri 
E Pr g0 > Tr0
g0 


gi 
BSi ∈8I


ri
r0
=
=
E{e−gi x }=µ[µ+x]−1
=
i6 =0








X


−α 
α

gi ri 
E exp −µTr0
gi 


ri 
BSi ∈8I


r0
i6 =0






 Y
r0 α 
E
E exp −µgi T
ri 
gi

ri

r0 

BSi ∈8I
i6 =0

 







  Y
r0 α −1
µ µ + µT
E E
r0 
ri 


ri


 
BSi ∈8I



i6 =0
VOLUME 8, 2020
M. Nemati et al.: RIS-Assisted Coverage Enhancement in mmWave Cellular Networks
Campbell’s theorem
=
n
E exp − 2πλI 5
r0
Z ∞ h r iα −1 0
1− 1+T
υdυ
.
υ
υ=r0
(56)
With changing the variable as u =
r0 , we have
υ2
and averaging over
2
r02 (T ) α
Pr [0o > T ] =
2 R∞
1
2
α
du
d r0 .
α
2
r0 =0 fro (ro ) exp −π λI r0 (T )
u=(T )− α 1+u 2
R∞
(57)
APPENDIX C
POWER-DENSITY
CONVERSION
p
Let r = x 2 + y2 and x, y ∈ 8 (homogeneous PPP) with
intensity of λ in the 2-dimensional
pEuclidean plane. Similarly,
let also consider a new r.v. R = X 2 + Y 2 where X , Y ∈ 8̃
with intensity of λ̃. Suppose constant values of P and α where
we have
h
i−α
1
R−α = Pr −α = (P)− α r
−α

v
2

u − α2 2
= u
x + (P)− α y2
. (63)
t(P)
| {z } | {z }
X
Y
Eventually, the closed form of (57) becomes
Pr [0o > T ]
λBS
2 R∞
=
λBS + λI T α
λ
λI = √BS
=
N
1+
√1 T
N
2
α
2
T−α
1
R∞
2
T−α
Therefore, we have
1
α
1+u 2
1
α
1+u 2
A
du
.
du
r2 =0 Pr(r0
> r2 |r2 )fr2 (r2 )dr2
Fr2 (r0 )
. (59)
Then, the denominator of (59) becomes
Fr2 (r0 ) = Pr [r2 < r0 ]
Z ∞ Z r0
=
fr2 (r2 ) dr2 fr0 (r0 )dr0
r2 =0
λRIS
=
.
λBS + λRIS
(61)
Eventually, by substituting (60) and (61) into (59), we have
2
Fr2 (R|r2 < r0 ) = 1 − e−π(λRIS +λBS )R .
(62)
Consequently, we have
2
fr2 (r|r2 < r0 ) = 2π(λRIS + λBS )re−π(λRIS +λBS )r .
VOLUME 8, 2020
P− α
1
2
(60)
The analysis is complete.
0
#{ x
.
y
λ̃ = det[A−1 ]λ = P α λ.
From (1) and (8), the numerator of (59) becomes
Z R h
i
λRIS
2
e−πλBS r2 fr2 (r2 )dr2 =
λ
BS + λRIS
r2 =0
h
i
2
× 1 − e−π(λRIS +λBS )R .
r0 =0
0
P
(64)
Consequently, profiting from mapping theorem, we can conclude 8 → 8̃ where
Since, the BSs and RISs are independently distributed in the
area and utilizing Bayes theorem, the CDF of being r2 < r0
is given by
Fr2 (R|r2 < r0 ) =
=
}|
− α1
(58)
APPENDIX B
DERIVATION OF fr2 (r |r2 < r0 )
RR
X
Y
z
"
(65)
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MAHYAR NEMATI (Member, IEEE) received the
B.S. degree in electrical engineering and telecommunications from the University of Tehran, Iran,
in 2015, and the M.S. degree in electrical, electronics, and cyber systems from Istanbul Medipol University, Turkey, in 2017. He is currently a Research
Assistant with the School of IT, Deakin University,
where he is involved in the field of wireless communications. His research interests include digital
communications, signal processing techniques at
the physical and medium access layer, multi-carrier schemes, waveform
design in wireless networks, the IoT, MTC, and URLLC.
VOLUME 8, 2020
JIHONG PARK (Member, IEEE) received the
B.S. and Ph.D. degrees from Yonsei University, Seoul, South Korea, in 2009 and 2016,
respectively. He was a Postdoctoral Researcher
with Aalborg University, Denmark, from 2016 to
2017, and the University of Oulu, Finland, from
2018 to 2019. He is currently a Lecturer with
the School of IT, Deakin University, Australia.
His research interests include ultra-dense/ultrareliable/mmWave system designs using stochastic
geometry, as well as distributed learning/control/ledger technologies and
their applications for beyond 5G/6G communication systems. He served
as a Conference/Workshop Program Committee Member for the IEEE
GLOBECOM, ICC, and WCNC, as well as for NeurIPS, ICML, and IJCAI.
He received the IEEE GLOBECOM Student Travel Grant in 2014, the IEEE
Seoul Section Student Paper Contest Bronze Prize in 2014, and the 6th
IDIS-ETNEWS (The Electronic Times) Paper Contest Award sponsored by
the Ministry of Science, ICT, and Future Planning of Korea. He is currently
an Associate Editor of Frontiers in Data Science for Communications,
a Review Editor of Frontiers in Aerial and Space Networks, and a Guest
Editor of Telecom (MDPI).
JINHO CHOI (Senior Member, IEEE) was born
in Seoul, South Korea. He received the B.E.
degree (magna cum laude) in electronics engineering from Sogang University, Seoul, in 1989,
and the M.S.E. and Ph.D. degrees in electrical
engineering from the Korea Advanced Institute of
Science and Technology (KAIST), in 1991 and
1994, respectively. He is currently a Professor with
the School of Information Technology, Deakin
University, Burwood, Australia. Prior to joining
Deakin in 2018, he was with Swansea University, U.K., as a Professor/Chair in Wireless, and the Gwangju Institute of Science and Technology
(GIST), South Korea, as a Professor. His research interests include the
Internet of Things (IoT), wireless communications, and statistical signal
processing. He authored two books published by Cambridge University
Press, in 2006 and 2010. He received the 1999 Best Paper Award for
Signal Processing from EURASIP and the 2009 Best Paper Award from
WPMC (Conference). He is currently an Editor of the IEEE TRANSACTIONS
ON COMMUNICATIONS and the IEEE WIRELESS COMMUNICATIONS LETTERS and a
Division Editor of Journal of Communications and Networks (JCN). He had
also served as an Associate Editor or Editor for other journals, including the
IEEE COMMUNICATIONS LETTERS, JCN, the IEEE TRANSACTIONS ON VEHICULAR
TECHNOLOGY, and ETRI Journal.
188185